Speed of sound and Mach number

Mach number

$$Ma = \frac{V}{a}$$

where $V$ is the flow velocity and $a$ is the local speed of sound. It measures the importance of compressibility: the ratio of inertia force to elastic force.

Speed of sound

Consider a small pressure wave moving at speed $a$ into still fluid of density $\rho$ and pressure $p$. In a frame fixed to the wave, the fluid approaches at speed $a$ and leaves at $a - dV$, with properties $\rho + d\rho$ and $p + dp$.

   still fluid         wave        disturbed fluid
   p, rho, V = 0  -->  |  a  |    p+dp, rho+drho, dV

Continuity: $\rho a = (\rho + d\rho)(a - dV)$. Neglecting the product $d\rho\,dV$:

$$dV = a\frac{d\rho}{\rho}$$

Momentum: $p - (p + dp) = \rho a\,[(a - dV) - a]$, so $dp = \rho a\,dV$.

Eliminating $dV$:

$$a^2 = \frac{dp}{d\rho} \quad \Rightarrow \quad a = \sqrt{\frac{dp}{d\rho}} = \sqrt{\frac{E_v}{\rho}}$$

where $E_v$ is the bulk modulus. The small wave is nearly reversible and adiabatic, so the process is isentropic: $p/\rho^k =$ constant, which gives $dp/d\rho = kp/\rho$. With $p = \rho RT$:

$$a = \sqrt{kRT}$$

For air, $k = 1.4$, $R = 287\ \text{J/kg K}$, so $a = 20.05\sqrt{T}$ m/s (343 m/s at 20 °C).

Classification of flow

Incompressible treatment

For $Ma < 0.3$ the density change in the flow is less than about 5%, so the flow can be treated as incompressible. Above this value compressibility effects must be included.